Frequency division multiplexing (FDM) is a multiplexing technique that has applications for both wireline and wireless applications. The former include, inter alia, applications to digital subscriber loop communications, and the latter include, inter alia, applications to WaveLan communications.
FDM in certain implementations is described in the discrete multitone (DMT) standard published by the International Telecommunications Union (ITU) under the publication number T1.413.
Very briefly, FDM operates by dividing a data stream into a temporal sequence of blocks. With reference to FIG. 1, the data in each block (the "information sequence") are represented, in frequency space, by a sequence of tones 10 (also referred to as carriers). The number of tones used varies among the various implementations, but is typically in the range 32-512. Information is encoded by assigning to each tone a complex amplitude (that is, a positive magnitude and a phase). The complex amplitudes that can be assigned are not arbitrary, and they do not vary continuously. Instead, they are typically drawn from a discrete set, sometimes referred to as a constellation, of points in the complex plane. (An illustrative such constellation is shown in FIG. 2.) Although the number of points in such a constellation may vary, a typical number of points lies in the range 2-512. In the case, e.g., of 256 points, each complex amplitude represents log.sub.2 256, or 8, bits of data. The sequence of amplitudes is referred to as the "signal sequence", and each amplitude in this sequence is referred to as a "signal element".
The transmitted signal (shown as element 20 in FIG. 1) is the Fourier transform of the sequence of tones. (Those skilled in the art will appreciate that what is precisely meant here is a real-valued reverse Fourier transform from the frequency domain to the time domain.) The receiver performs, in essence, another Fourier transform back into frequency space to recover the signal sequence. At the receiver, knowledge concerning the mapping between the information sequence and the signal sequence is used to recover the information sequence.
Those familiar with oscillatory phenomena will appreciate that when tones, with associated phase differences, are superimposed, constructive interference often leads to the emergence of peaks that extend to a significant height above the average amplitude of the combined waveform. When a transmitted signal exhibits this property, it is often useful to characterize the signal by its peak-to-average power ratio. It should be noted in this regard that the average transmitted power is directly related to the rate at which information can be communicated. That is, higher average power implies higher potential data-communication rate. The term "average" refers here to the statistical, or ensemble, average over all signals. The term "peak", on the other hand, refers to a particular sequence.
For several reasons, it is desirable to limit the peak-to-average power ratio of a transmitted signal. In many instances, there are standards and regulations that impose a limit on this quantity. More fundamentally, the last amplification stage of the transmitter may be saturated by amplitude peaks in its input signal, resulting in clipping of the transmitted waveform and consequent errors in data transmission. In principle, an amplifier can be designed to handle essentially any given peak-to-average power ratio encountered in practice. However, the cost of the amplifier and also the power consumption of the amplifier increase as the maximum acceptable peak power increases. As a consequence, economic considerations militate for measures designed to minimize the probability that a peak will appear that cannot be transmitted without distortion. Although the greatest acceptable value for this probability depends on the specific application and on other factors, a typical value is 10.sup.-5.
One conventional approach to this problem is to limit the permitted signals to only those signals that have an acceptably low peak power. This approach is difficult to put into practice because it is difficult to identify a suitable set of signals in the frequency domain. Moreover, decreasing the population of available signals decreases the number of bits carried by each signal element. Because a substantial fraction of signals are typically eliminated by this approach, the transmitted data rate is significantly reduced.